Note that because of the truncated mode expansion (REF ), the mode number \(n\)  on \(H^k_n\)  cannot be lowered indefinitely by acting with \(\bar{L}_{-1}\) . \(H^k_{\frac{k-4}{2}}\)  is both a primary and a descendant, and therefore null in an appropriately-defined \({\rm SL}(2,\mathbb {R})_R\)  covariant norm for celestial conformal field theory [1]}, [2]}, [3]}.The importance of global conformal primary descendants of conformally soft celestial operators was first pointed out in [4]}, [5]}. This is in contrast with the \(\widehat{\rm w}^q_n\)  modes, which form a closed algebra with the \({\rm SL} (2, \mathbb {R})_R\)  generators without any primary descendants or null states.
